Araştırma Makalesi

Yıl 2024,
Cilt: 12 Sayı: 3, 101 - 112
### Öz

### Anahtar Kelimeler

### Kaynakça

In this study, we investigate Smarandache curves constructed by a space curve with a modified orthogonal frame. Firstly, the relations between the Frenet frame and the modified orthogonal frame are summarized. Later, the Smarandache curves based on the modified orthogonal frame are obtained. Finally, the tangent, normal, binormal vectors and the curvatures of the Smarandache curves are determined. A special curve known as the Gerono lemniscate curve whose curvature is not differentiable, the principal normal and binormal vectors are discontinuous at zero point is considered as an example, and the Smarandache curves of this curve are obtained by the aid of its modified orthogonal frame, and their graphics are given.

Gerono lemniscate curve Modified orthogonal frame Smarandache curves

- [1] Ali, A. T.: Special Smarandache curves in the Euclidean space. International Journal of Mathematical Combinatorics. 2, 30-36 (2010).
- [2] Okuyucu, O. Z., Değirmen, Ç, Yıldız, Ö. G.: Smarandache curves in three dimensional Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 68(1), 1175-1185 (2019).
- [3] Taşköprü, K., Tosun, M.: Smarandache curves on S2. Boletim da Sociedade Paranaense de Matematica. 32(1), 51-59 (2014).
- [4] Gürses, B. N., Bektaş, Ö, Yüce, S.: Special Smarandache curves in R^3_1. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 65(2), 143-160 (2016).
- [5] Turgut, M., Yılmaz, S.: Smarandache curves in Minkowski space-time. International Journal of Mathematical Combinatorics. 3, 51-55 (2008).
- [6] Ergut, M., Yılmaz, S., Ünlütürk, Y.: Isotropic Smarandache curves in the complex 4-space. Honam Mathematical Journal. 40(1), 47–59 (2018).
- [7] Solouma, E. M., Mahmoud,W. M.: On spacelike equiform-Bishop Smarandache curves on S2 1 . Journal of the Egyptian Mathematical Society. 27(1), 7-17 (2019).
- [8] Solouma, E. M.: Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF. Bulletin of Mathematical Analysis and Applications. 13(4), 14-30 (2021).
- [9] Şenyurt, S., Ayvacı, K. H., Canlı, D.: Smarandache curves according to Flc-frame in Euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences. 4(1), 16-30 (2023).
- [10] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame. Gümü¸shane University journal of Science. 10(1), 251-260 (2020).
- [11] Şenyurt, S., Eren, K.: Smarandache curves of spacelike Salkowski curve with a spacelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(Special Issue -I), 7-17 (2020).
- [12] Şenyurt, S., Eren, K.: Some Smarandache curves constructed by a spacelike Salkowski curve with timelike principal normal. Punjab University Journal of Mathematics. 53(9), 679-690 (2021).
- [13] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a timelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(2), 404-416 (2020).
- [14] Özen K. E., Tosun, M.: Trajectories generated by special Smarandache curves according to positional adapted frame. KMU Journal of Engineering and Natural Sciences. 3(1), 15-23 (2021).
- [15] Özen K. E., Tosun, M., Avcı, K.: Type 2-positional adapted frame and its application to Tzitzeica and Smarandache curves. Karatekin University Journal of Science. 1(1), 42-53 (2022).
- [16] Sasai, T.: The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Mathematical Journal. 36, 17–24 (1984).
- [17] Bükcü, B., Karacan, M. K.: On the modified orthogonal frame with curvature and torsion in 3-space. Math. Sci. Appl. E-Notes. 4, 184–188 (2016).
- [18] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 2010.

Yıl 2024,
Cilt: 12 Sayı: 3, 101 - 112
### Öz

### Kaynakça

- [1] Ali, A. T.: Special Smarandache curves in the Euclidean space. International Journal of Mathematical Combinatorics. 2, 30-36 (2010).
- [2] Okuyucu, O. Z., Değirmen, Ç, Yıldız, Ö. G.: Smarandache curves in three dimensional Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 68(1), 1175-1185 (2019).
- [3] Taşköprü, K., Tosun, M.: Smarandache curves on S2. Boletim da Sociedade Paranaense de Matematica. 32(1), 51-59 (2014).
- [4] Gürses, B. N., Bektaş, Ö, Yüce, S.: Special Smarandache curves in R^3_1. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 65(2), 143-160 (2016).
- [5] Turgut, M., Yılmaz, S.: Smarandache curves in Minkowski space-time. International Journal of Mathematical Combinatorics. 3, 51-55 (2008).
- [6] Ergut, M., Yılmaz, S., Ünlütürk, Y.: Isotropic Smarandache curves in the complex 4-space. Honam Mathematical Journal. 40(1), 47–59 (2018).
- [7] Solouma, E. M., Mahmoud,W. M.: On spacelike equiform-Bishop Smarandache curves on S2 1 . Journal of the Egyptian Mathematical Society. 27(1), 7-17 (2019).
- [8] Solouma, E. M.: Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF. Bulletin of Mathematical Analysis and Applications. 13(4), 14-30 (2021).
- [9] Şenyurt, S., Ayvacı, K. H., Canlı, D.: Smarandache curves according to Flc-frame in Euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences. 4(1), 16-30 (2023).
- [10] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame. Gümü¸shane University journal of Science. 10(1), 251-260 (2020).
- [11] Şenyurt, S., Eren, K.: Smarandache curves of spacelike Salkowski curve with a spacelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(Special Issue -I), 7-17 (2020).
- [12] Şenyurt, S., Eren, K.: Some Smarandache curves constructed by a spacelike Salkowski curve with timelike principal normal. Punjab University Journal of Mathematics. 53(9), 679-690 (2021).
- [13] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a timelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(2), 404-416 (2020).
- [14] Özen K. E., Tosun, M.: Trajectories generated by special Smarandache curves according to positional adapted frame. KMU Journal of Engineering and Natural Sciences. 3(1), 15-23 (2021).
- [15] Özen K. E., Tosun, M., Avcı, K.: Type 2-positional adapted frame and its application to Tzitzeica and Smarandache curves. Karatekin University Journal of Science. 1(1), 42-53 (2022).
- [16] Sasai, T.: The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Mathematical Journal. 36, 17–24 (1984).
- [17] Bükcü, B., Karacan, M. K.: On the modified orthogonal frame with curvature and torsion in 3-space. Math. Sci. Appl. E-Notes. 4, 184–188 (2016).
- [18] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 2010.

Toplam 18 adet kaynakça vardır.

Birincil Dil | İngilizce |
---|---|

Konular | Uygulamalı Matematik (Diğer) |

Bölüm | Articles |

Yazarlar | |

Erken Görünüm Tarihi | 14 Nisan 2024 |

Yayımlanma Tarihi | |

Gönderilme Tarihi | 24 Aralık 2023 |

Kabul Tarihi | 13 Mart 2024 |

Yayımlandığı Sayı | Yıl 2024 Cilt: 12 Sayı: 3 |